NavList:

Johnson was an interesting chap - born in Cambridge (UK) c.1828 and a graduate of Gonville & Caius College in 1853, he worked as an instructor in the Royal Navy (initially as an embarked schoolmaster for the education of midshipmen) from 1855-1887. By 1877 he was the senior instructor in the British Royal Navy’s cadet training ship, HMS Britannia, and taught both the future Edward VIII and George V. He was a prolific author of instructional books in the mathematics of navigation - I’ve found about 14 separate titles by him, and WorldCat lists a total of 36 though many seem to be updates/improvements and retitled versions of earlier works, at least some of which were translated into French and Turkish(!). His first seems to have been An introduction to plane and spherical trigonometry, 1865, which went through nine editions, the ninth of 1899 changing the title to The elements of plane and spherical trigonometry.

Some of his works were obviously major stand-bys, his On finding the latitude and longitude in cloudy weather and at other times, went into its 40th edition in 1929 (20 years after Johnson’s death) and, as far as I can work out, may well have been the final iteration of the book you are looking for, WorldCat (which works on titles) begins its listing with the 33rd, 1912 edition but one infers a change of title at that point. The 36th edition is on the Internet Archive - see https://archive.org/details/onfindinglatitud00johnrich. 

Stephen D